Abstract— Offshore wind and tidal current are of the most
common energy resources for generating electricity in the near
future because of the oil problems (crises and pollution). The
dynamic model of the offshore wind and tidal current is very
important topic for dealing with these renewable energies. This
paper describes the overall dynamic models of offshore wind
and tidal current turbine using three different types of
generators (doubly fed induction generator (DFIG), squirrel
cage induction generator (SCIG) and direct drive permanent
magnet synchronous generator (DDPMSG)). The state space
for all types of the generators are concluded. All models are
validated using a common property of the generator for the
validation.
Index Terms— Offshore wind, Tidal current, Direct drive
permanent magnet synchronous generator (DDPMSG), Doubly
fed induction generator (DFIG).
NOMENCLATURE
Vtide Tidal current speeds.
Vnt Neap tide speed.
Vst Spring tide speed.
Pts Tidal in-stream power.
Density of the water (1025 kg/m3)
A Cross-sectional area perpendicular to the flow
direction.
Tm Mechanical torque applied to the turbine.
A Cross-sectional area perpendicular to the flow
direction.
Cp Marine turbine blade design constant in the
range of 0.35-0.5.
ωs ,ωr ,
ωt
Stator, rotor electrical angular velocities, and
turbine speed at hub height upstream the rotor.
Te Electrical torque of the generator.
Ds Shaft stiffness damping.
Ht ,Hg Turbine and generator inertia constants.
Ks Shaft stiffness coefficient.
Ɵt ,Ɵr Turbine and generator rotor angles.
β Tidal turbine pitch angle.
S Rotor slip.
d, q Indices for the direct and quadrature axis
components.
s, r Indices of the stator and the rotor.
Manuscript received June 21, 2013; revised July 13, 2013. This work
was supported in part by the Egyptian Governoment and Dalhousie
University. Hamed H. H. Aly was with Zagazig University, Zagazig, Egypt. He is
now with the Department of Electrical & Computer Engineering at
Dalhousie University, NS, B3H 4R2, Canada. (e-mail: [email protected]). M. E. El-Hawary is a professor at the Department of Electrical and
Computer Engineering, Dalhousie University, Halifax, NS, B3H 4R2,
Canada (e-mail: [email protected]).
v, R, i,
ψ
Voltage, resistance, current, and flux linkage
of the generator.
, Coefficients for the proportional-integral
controller of the pitch controller.
Pg, PDC Active power of the AC terminal at the grid
side converter and DC link power
respectively.
vDg , vQg D and Q axis voltages of the grid side
converter.
iDg ,iQg D and Q axis currents of the grid side
converter.
C Capacitance of the capacitor.
vDC , iDC Voltage and current of the capacitor.
Kp1, Kp2,
Kp3
Proportional controller constants for the
generator side converter controller
Ki1, Ki2
,Ki3
Integral controller constants for the generator
side converter controller.
iDg, iQg D and Q axis grid currents.
vDg, vQg D and Q axis grid voltages.
Kp4, Kp5,
Kp6
Proportional controller constants for the grid
side converter.
Ki4, Ki5,
Ki6
Integral controller constants for the grid side
converter.
Xc Grid side smoothing reactance.
State variable
I. INTRODUCTION
ind is hardly predictable source of energy. Tidal
current turbines extract electric energy from the
water. Tidal currents are fluctuating, intermittent but
a predictable source of energy compared to wind. Its use is
very effective as it relies on the same technologies used in
wind turbines. The electrical-side layout and modeling
approaches used in tidal in-stream systems are similar to
those used for wind and offshore wind systems. The speed
of water currents is lower than wind speed, while the water
density is higher than the air density and as a result wind
turbines operate at higher rotational speeds and lower torque
than tidal in-stream turbines which operate at lower
rotational speed and high torque [1-5].
The easier predictability of the tidal in-stream energy
resource makes it easier to integrate in an electric power
grid. Recognizing that future ocean energy resources are
available far from load centers and in areas with limited grid
capacity will result in challenges and technical limitations.
With the growing penetration of tidal current energy into the
electric power grid system, it is very important to study the
impact of tidal current turbines on the stability of the power
system grid and to do that we should model the overall
Modeling of Offshore Wind and Tidal Current
Turbines for Stability Analysis
Hamed H. H. Aly and M. E. El-Hawary
W
Proceedings of the World Congress on Engineering and Computer Science 2013 Vol I WCECS 2013, 23-25 October, 2013, San Francisco, USA
ISBN: 978-988-19252-3-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCECS 2013
system. The model of the ocean energy system consists of
three stages. The first stage contains the fluid mechanical
process. The second stage consists of the mechanical
conversion and depends on the relative motion between
bodies. This motion may be mechanical transmission and
then using mechanical gears or may be depending on the
hydraulic pumps and hydraulic motors. The third stage
consists of the electromechanical conversion to the electrical
grid [4-9].
II. OFFSHORE WIND AND TIDAL CURRENT MODEL USING IG
A. The Speed Signal Resource Model for the Tidal
Current and Offshore Wind
The tidal current speed may be expressed as a function of
the spring tide speed, neap tide speed and tides coefficient.
Paper (10) proposed different algorithm models for the tidal
current speed.
The Wind Speed Signals Model (vw) consists of four
components related to the mean wind speed (vmw), the wind
speed ramp (vrw) (which is considered as the steady increase
in the mean wind speed), the wind speed gust (vgw), and the
turbulence (vtw). vw = vmw + vgw + vrw + vtw
The mean wind speed is a constant; a simple ramp
function will be used for ramp component (characterized by
the amplitude of the wind speed ramp (Ar (m/s)), the starting
time (Tsr), and the ending time (Ter)). The wind speed gust
component is characterized by the amplitude of the wind
speed gust (Ag (m/s)), the starting time (Tsg), and the ending
time (Teg). The wind speed gust may be expressed as a
sinusoidal function. The most used models are given by:
vgw= Ag(1-cos(2(t/Dg-Tsg/Dg))) Tsg ≤ t≤ Teg
vgw =0 t< Tsg or t >Teg
Dg= Teg - Tsg
A triangular wave is used to represent the turbulence
function which has adjustable frequency and amplitude [11].
B. The Rotor Model
For the offshore wind the rotor model represents the
conversion of kinetic energy to mechanical energy. The
wind turbine is characterized by Cp (wind power
coefficient), λ (tip speed ratio), and β (pitch angle). λ =
ωtR/vw , where R is the blade length in m, vw is the wind
speed in m/s, and ωt is the wind turbine rotational speed in
rad/sec. CP-λ-β curves are manufacturer-dependent but there
is an approximate relation expressed as
,
Cf is the wind turbine blade design constant. The rotor
model may be represented by using the equation of the
power extracted from the wind (Pw = ) [11].
For the tidal current turbines the rotor model may be
represented by using the equation of the power extracted
from the wind (Pw = ),
The power (Pts) may be found using: Pts = ½ A(Vtide)3. The
turbine harnesses a fraction of this power, hence the power
output may be expressed as: Pt = ½ CpA(Vtide)3. The power
output is proportional to the cube of the velocity. The
velocity at the bottom of the channel is lower than at the
water column above seabed. The mechanical torque applied
to the turbine (Tm) can be expressed as [4-9]:
The shaft system for tidal current and offshore turbines may
be represented by a two mass system one for the turbine and
the other for the generator as shown:
2Ht
= Tt – Ks(Ɵr-Ɵt) – Ds (ωr- ωt)
(2)
2Hg
= Te - Ks(Ɵr-Ɵt) – Ds (ωr- ωt)
(3)
Ɵtr = Ɵr-Ɵt (4)
Ɵ
= ωr- ωt
(5)
There is a ratio for the torsion angles, damping and stiffness
that need to be considered when one adds a gear box as all
above calculations must be referred to the generator side and
calculated as : a=
,
,
,
a
2
,
=a2
. The same model used for the offshore wind is
used for tidal in-stream turbines; however, there is a number
of differences in the design and operation of marine turbines
due to the changes in force loadings, immersion depth, and
different stall characteristics.
C. Dynamic Model of DFIG
The DFIG model is developed using a synchronously
rotating d-q reference frame with the direct-axis oriented
along the stator flux position. The reference frame rotates at
the same speed as the stator voltage. The stator and rotor
active and reactive power are given by [4-10]:
Ps=3/2( vds ids + vqs iqs ) , Pr= 3/2(vdr idr + vqr iqr ) (6)
Pg= Ps + Pr (7)
Qs=3/2(vqs ids - vds iqs ) , Qr=3/2 (vqr idr - vdr iqr ) (8)
The model of the DFIG can be described as:
vds = -Rs ids - ωs ψqs +
(9)
vqs = -Rs iqs + ωs ψds +
(10)
vdr = -Rr idr - s ωs ψqr +
(11)
vqr = -Rr iqr + s ωs ψdr +
(12)
Ψds = -Lss ids - Lm idr , Ψqs = -Lss iqs - Lm iqr (13)
Ψdr = -Lrr idr - Lm ids , Ψqr = - Lrr iqr - Lm iqs (14)
s =( ωs –ωr )/ωs (15)
= -ωs
(16)
Where, Lss = Ls + Lm, Lrr = Lr + Lm, Ls, Lr and Lm are
the stator leakage, rotor leakage and mutual inductances,
respectively. The previous model may be reduced by
neglecting stator transients and is described as follows: vds = -Rs ids + Xˈ iqs + ed (17)
vqs = -Rs iqs - Xˈ ids + eq (18)
(19)
(20)
The components of the voltage behind the transient are
and
. The stator reactance
X = ωs Lss = Xs +Xm, and the stator transient reactance Xˈ =
ωs (Lss – Lm2/Lrr) = Xs + (Xr Xm)/ (Xr + Xm). The transient
open circuit time constant is To = Lrr/Rr = (Lr +Lm)/Rr, and
the electrical torque is Te= (idsiqr-iqsidr)Xm/ . Figure (1)
shows d and q – axis equivalent circuit of DFIG.
(1)
Proceedings of the World Congress on Engineering and Computer Science 2013 Vol I WCECS 2013, 23-25 October, 2013, San Francisco, USA
ISBN: 978-988-19252-3-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCECS 2013
d – axis equivalent circuit of DFIG
q – axis equivalent circuit of DFIG
Figure (1) d and q – axis equivalent circuit of DFIG.
d – axis equivalent circuit of SCIG
q – axis equivalent circuit of SCIG
Figure (2) d and q – axis equivalent circuit of FSIG
D. Dynamic Model of SCIG
In the SCIG the rotor is short circuited and so, the stator
voltages are the same as the DFIG. In the SCIG there are
capacitors to provide the induction generator magnetizing
current and for compensation. Figure (2) d and q – axis
equivalent circuit of FSIG.
E. The Pitch Controller Model
The pitch controller is used to adjust the tidal current turbine
to achieve a high speed magnitude. This may be represented
by a PI controller with a transfer function +
.
Figure (3) Pitch angle control block
β=( +
)ωt (21)
=
+ ωt
(22)
F. The Converter Model
A converter feeds or takes power from the rotor circuit and
gives a variable speed (a partial scale power converter used).
The rotor side of the DFIG is connected to the grid via a
back to back converter. The converter at the side connected
to the grid is called supply side converter (SSC) or grid side
converter (GSC) while the converter connected to the rotor
is the rotor side converter (RSC). The RSC operates in the
stator flux reference frame. The direct axis component of
the rotor current acts in the same way as the field current as
in the synchronous generator and thus controls the reactive
power change. The quadrature component of the rotor
current is used to control the speed by controlling the torque
and the active power change. Thus the RSC governs both
the stator-side active and reactive powers independently.
The GSC operates in the stator voltage reference frame. The
d-axis current of the GSC controls the DC link voltage to a
constant level, and the q-axis current is used for reactive
power control. The GSC is used to supply or draw power
from the grid according to the speed of the machine. If the
speed is higher than synchronous speed it supplies power,
otherwise it draws power from the grid but its main
objective is to keep dc-link voltage constant regardless of
the magnitude and direction of rotor power The back to back
converter using a DC link is shown in Figure (4). The
balanced power equation is given by [13, 14]:
Pr=Pg+PDC (23)
PDC=vDCiDC=-CvDC
, Pg=vDgiDg+vQgiQg
(24)
CvDC
= vDgiDg+vQgiQg - vdr idr - vqr iqr (25)
Figure (4) Back to back converter
G. Rotor Side Converter Controller Model
Figure (5) Generator side converter controller
The rotor side converter controller used here is represented
by four states ( , , and ), is related to the
difference between the generated power of the stator and the
reference power that is required at a certain time, is
related to the difference between the quadrature axis
generator rotor current and the reference current that is
required at a certain time, is related to the difference
between the stator terminal voltage and the reference
Lm
+
ωt β
Proceedings of the World Congress on Engineering and Computer Science 2013 Vol I WCECS 2013, 23-25 October, 2013, San Francisco, USA
ISBN: 978-988-19252-3-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCECS 2013
voltage that is required at a certain time, and is related to
the difference between the direct axis generator rotor current
and the reference current that is required at a certain time
[13, 14]. Figure (5) shows the rotor side converter
controller. This is described by equations (26-34).
= Pref -Ps (26)
= - Ki1/ Kp1x1 + 1/ Kp1 iqr_ref (27)
= iqr_ref -iqr (28)
= Kp1 + Ki1 x1 - iqr (29)
=- / x2 +1/ vqr - ωsLm/ ids -ωsLrr/ idr +
(Lm/ ) idsωr +(Lrr/ )idrωr
(30)
= vs_ref - vs (31)
= - Ki3/ Kp3x3 + 1/ Kp3 idr_ref (32)
= idr_ref - idr (33)
=- / x4 +1/ vdr - ωsLm/ iqs -ωsLrr/ iqr +
(Lm/ ) iqsωr +(Lrr/ )iqrωr
(34)
H. Grid Side Converter Controller Model
The grid side converter controller used here is represented
by four states ( , , and ), x5 is related to the
difference between the DC voltage and the reference DC
voltage required at a certain time, x6 related to the
difference between the grid terminal voltage and the
reference terminal voltage required at a certain time, x6 is a
combination of x5 and direct axis grid current and x8 is a
combination of x6 and quadrature axis grid current as shown
in equations (35-40). Figure (6) shows the grid side
converter controller.
Figure (6) Grid side converter controller
= vDC_ref -vDC (35) = Kp4 +Ki4x5-iDg (36) = vt_ref –vt (37) = Kp6 +Ki6x7-iQg (38) vDg=Kp5 +Ki5x6+XciQg (39) vQg=Kp5 +Ki5x8- XciDg (40)
III. TIDAL CURRENT TURBINE MODEL USING DDPMSG
A. The Dynamic Modeling of The DDPMSG
The DDPMSG can be modeled as the following [3, 13]:
vds =-Rs × ids - ωs × ψqs +
(41)
vqs =-Rs × iqs + ωs × ψds +
(42)
The flux linkages and the torque can be expressed as:
Ψds = - Ld × ids + ψf (43)
Ψqs = -Lq × iqs
Te= (3/2)p iqs ((Ld-Lq) ids + ψf )
(44)
(45)
Ld, and Lq are the direct and quadrature inductances of
the stator. Ψf is the excitation field linkage, and p is the
number of pair poles. Figure (7) shows the d-q axis
component of the DDPMSG. In this paper for simplicity we
will assume that Ld=Lq=Ls , and so the generator model can
be rewritten in a state space representation as:
Ls
- vds -Rs × ids + Ls× ωs × (46)
Ls
-vqs -Rs × iqs - Ls× ωs × +ω × ψf (47)
The converters models used for the DFIG are the same
converters that used for the DDPMSG keeping in mind that
a full scale power converter is used.
Figure (7) d-q axis component of PMSG
B. The Generator Side Converter Controller Model for
DDPMSG
The generator side converter controller used here is
represented by two states only (x1, and x2), x1 related to the
difference between the generated power and the reference
power that required at a certain time and x2 related to the
difference between the direct axis generator current and the
reference current that required at a certain time [13-15].
Figure (8) shows the generator side converter controller
described by:
=Ps-Pref
=ids-ids_ref
(48)
(49)
vqs=Kp1 +Ki1x1-Lswids
vds=Kp2 +Ki2x2+Lswiqs
(50)
(51)
Where: Kp1, Kp2, represent the proportional controller
constants and Ki1, Ki2 represent the integral controller
constants for the generator side converter controller.
Figure (8) Generator side converter controller for DDPMSG
C. The Grid Side Converter Controller Model for
DDPMSG
The grid side converter controller used here is represented
by four states (x3, x4, x5, x6), x3 related to the difference
between the DC voltage and the reference DC voltage that
required at a certain time, x5 related to the difference
between the terminal voltage and the reference terminal
voltage that required at a certain time, x4 is a combination of
x3 and direct axis grid current and x6 is a combination of x4
Proceedings of the World Congress on Engineering and Computer Science 2013 Vol I WCECS 2013, 23-25 October, 2013, San Francisco, USA
ISBN: 978-988-19252-3-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCECS 2013
and quadrature axis grid current. Figure (9) shows the grid
side converter controller.
Figure (9) Grid side converter controller for DDPMSG
=vsDC_ref-vDC
=Kp3 +Ki3x3-iDg
(52)
(53)
=vt_ref–vt
=Kp4 +Ki4x5-iQg
vDg=Kp5 +Ki5x4+XciQg
vQg=Kp5 +Ki5x6-XciDg
(54)
(55)
(56)
(57)
Where: VDC is the DC link voltage, iDg, iQg are the D and Q
axis grid currents, vDg, vQg are the D and Q axis grid
voltages, Kp3, Kp4, Kp5 represent the proportional controller
constants, Xc is the grid side smoothing reactance, and Ki3,
Ki4, Ki5 represent the integral controller constants for the grid
side converter.
Figure (10) The relation between different values of the rotor resistance and
the eigenvalues of the changed mode in case of DFIG.
IV. VALIDATION OF THE MODELS
As the value of the resistance or inductance changes the
stability degree will change. Figure (10) shows the relation
between different values of the rotor resistance and the
eigenvalues of the changed mode in case of DFIG. Figure
(11) shows the relation between different values of the stator
resistance and the eigenvalues of the changed mode in case
of DDPMSG. Figure (12) shows the relation between
different values of the rotor inductance and the eigenvalues
of the changed mode in case of DFIG. Figure (13) shows the
relation between different values of the stator inductance
and the eigenvalues of the changed mode in case of
DDPMSG. As the resistance value of the rotor increase, the
stability degree will increase and as the inductance value
increase as the stability will decrease because of increasing
the delay time.
Figure (11) The relation between different values of the stator resistance and the eigenvalues of the changed mode in case of DDPMSG.
Figure (12) The relation between different values of the rotor inductance
and the eigenvalues of the changed mode in case of DFIG.
Figure (13) The relation between different values of the stator self inductance and the eigenvalues of the changed mode in case of DDPMSG.
V. CONCLUSION
The use of offshore wind and tidal current as a
renewable source of energy is very effective as it relies on
similar technologies. The overall dynamic system of the
offshore and tidal current turbine based on three different
types of generators for a single machine infinite bus system
has been modeled. The converter used has been modelled.
Different controller models are also discussed for different
machines. The state space representation of the overall
system is concluded. All models are validated using a
common property of the machine for the validation
(changing the degree of the system stability by changing of
the value of the resistance and the inductance). If the
resistance value of different generators increases, the
stability degree will increase and if the inductance value of
the machine increases, the stability degree will decrease.
The tidal current energy is more predictable compared to
offshore wind energy. It is beneficial to use a hybrid of
offshore wind and tidal current turbine.
Proceedings of the World Congress on Engineering and Computer Science 2013 Vol I WCECS 2013, 23-25 October, 2013, San Francisco, USA
ISBN: 978-988-19252-3-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCECS 2013
ACKNOWLEDGMENT
The first author would like to thank the Egyptian
Government for supporting this work.
REFERENCES
[1] ”Tidal Stream” Available online (January 2011),
http://www.tidalstream.co.uk/html/background.html [2] ”Tidal Currents” Available online (April 2011),
http://science.howstuffworks.com/environmental/earth/oceanography/
ocean-current4.htm [3] Hamed H. H. Aly, and M. E. El-Hawary “State of the Art for Tidal
Currents Electrical Energy Resources”, 24th Annual Canadian IEEE
Conference on Electrical and Computer Engineering, Niagara Falls, Ontario, Canada, 2011.
[4] Hamed H. Aly, and M. E. El-Hawary “An Overview of Offshore
Wind Electrical Energy Systems” 23rd Annual Canadian IEEE Conference on Electrical and Computer Engineering, Calgary,
Alberta, Canada, May 2-5, 2010.
[5] Marcus V. A. Nunes, J. A. Peças Lopes, Hans Helmut, Ubiratan H.
Bezerra, and Rogério G. “Influence of the Variable-Speed Wind
Generators in Transient Stability Margin of the Conventional
Generators Integrated in Electrical Grids” IEEE Transactions on Energy Conversion, 2004.
[6] J.G. Slootweg, H. Polinder and W.L. Kling“Dynamic Modeling of a
Wind Turbine with Doubly Fed Induction Generator” IEEE Power Engineering Society Summer Meeting, 2001.
[7] Janaka B. Ekanayake, Lee Holdsworth, XueGuang Wu, and Nicholas Jenkins” Dynamic Modeling of Doubly Fed Induction Generator
Wind Turbines” IEEE Transactions on Power Systems, Vol. 18, No.
2, May 2003. [8] M.J.Khan, G. Bhuyan, A. Moshref, K. Morison, “An Assessment of
Variable Characteristics of the Pacific Northwest Regions Wave and
Tidal Current Power Resources, and their Interaction with Electricity Demand & Implications for Large Scale Development Scenarios for
the Region,” Tech. Rep. 17485-21-00 (Rep 3), Jan. 2008.
[9] Lucian Mihet-Popa, Frede Blaabjerg, and Ion Boldea,” Wind Turbine Generator Modeling and Simulation Where Rotational Speed is the
Controlled Variable” IEEE Transactions On Industry Applications,
Vol. 58, No. 1, January/February 2004. [10] Hamed H. H. Aly, M. E. El-Hawary “A Proposed Algorithms for
Tidal in-Stream Speed Model” American Journal of Energy
Engineering. Vol. 1, No. 1, 2013, pp. 1-10. [11] Hamed H. H. Aly, M. E. El-Hawary. The Current Status of Wind and
Tidal in-Stream Electric Energy Resources, American Journal of
Electrical Power and Energy Systems. Vol. 2, No. 2, 2013, pp. 23-40. [12] Yazhou Lei, Alan Mullane, Gordon Lightbody, and Robert Yacamini
“Modeling of the Wind Turbine with a Doubly Fed Induction
Generator for Grid Integration Studies” IEEE Transaction on Energy Conversion, Vol. 28, No. 1, March 2006.
[13] F. Wu, X.-P. Zhang, and P. Ju “Small signal stability analysis and
control of the wind turbine with the direct-drive permanent magnet generator integrated to the grid” Journal of Electric Power and
Engineering Research, 2009.
[14] F. Wu, X.-P. Zhang, and P. Ju “Small signal stability analysis and optimal control of a wind turbine with doubly fed induction
generator” IET Journal of Generation, Transmission and Distribution,
2007. [15] Hamed H. Aly “Forecasting, Modeling, and Control of Tidal currents
Electrical Energy Systems”PhD thesis, Halifax, Canada. 2012.
Proceedings of the World Congress on Engineering and Computer Science 2013 Vol I WCECS 2013, 23-25 October, 2013, San Francisco, USA
ISBN: 978-988-19252-3-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCECS 2013
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